CW Complex - Inductive Definition of CW-complexes

Inductive Definition of CW-complexes

If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The n-skeleton of a CW complex is the union of the cells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex, called a subcomplex. Thus the n-skeleton is the largest subcomplex of dimension n or less.

A CW complex is often constructed by defining its skeleta inductively. Begin by taking the 0-skeleton to be a discrete space. Next, attach 1-cells to the 0-skeleton. Here, the 1-cells are attached to points of the 0-skeleton via some continuous map from unit 0-sphere, that is, . Define the 1-skeleton to be the identification space obtained from the union of the 0-skeleton, 1-cells, and the identification of points of boundary of 1-cells by assigning an identification mapping from the boundary of the 1-cells into the 1-cells. In general, given the (n − 1)-skeleton and a collection of (abstract) closed n-cells, as above, the n-cells are attached to the (n − 1)-skeleton by some continuous mapping from, and making an identification (equivalence relation) by specifying maps from the boundary of each n-cell into the (n − 1)-skeleton. The n-skeleton is the identification space obtained from the union of the (n − 1)-skeleton and the closed n-cells by identifying each point in the boundary of an n-cell with its image.

Up to isomorphism every n-dimensional complex can be obtained from its (n − 1)-skeleton in this sense, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the direct limit of the skeleta: a set is closed in X if and only if it meets each skeleton in a closed set.